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Equilibration of Multitime Quantum Processes in Finite Time Intervals

by Neil Dowling, Pedro Figueroa-Romero, Felix A. Pollock, Philipp Strasberg, Kavan Modi

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Neil Dowling
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-08
Date submitted: 2023-03-17 02:52
Submitted by: Dowling, Neil
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


A generic non-integrable (unitary) out-of-equilibrium quantum process, when interrogated across many times, is shown to yield the same statistics as an (non-unitary) equilibrated process. In particular, using the tools of quantum stochastic processes, we prove that under loose assumptions, quantum processes equilibrate within finite time intervals. Sufficient conditions for this to occur are that multitime observables are coarse grained in both space and time, and that the initial state overlaps with many different energy eigenstates. These results help bridge the gap between (unitary) quantum and (non-unitary) statistical physics, i.e., when all multitime properties and correlations are well approximated by stationary quantities, which includes non-Markovianity and temporal entanglement. We discuss implications of this result for the emergence of classical stochastic processes from multitime measurements of an underlying genuinely quantum system.

Author comments upon resubmission

We thank the Referees for carefully reviewing the manuscript and providing useful suggestions and feedback. We believe that this has significantly improved the paper.

In particular, we have: fixed a range of insufficient explanations and typos, more elegantly explained the relevance and impact of the results, added further comparison to related work and the state-of-the-art, detailed how this work fits into and contributes in a unique way to the overarching research program, and added more critical discussion of the results.

We hope the Editor and Referees agree that the changes, as detailed below, make this work a suitable candidate for publication in SciPost Physics Core.

List of changes

- New abstract
- Updated paragraph 2 of introduction to include more background information on different approaches to the question of statistical mechanics from pure state quantum mechanics.
- Added more citations from across a range of research areas in quantum thermalization/equilibration, largely in the second paragraph on page 1 (Refs. [10-18]) and below Eq. (29) (Refs. [13,14,58,59,60]).
- Updated paragraph 3 of introduction to explain that multitime equilibration implies a robustness to perturbation.
- Added paragraph 4 to introduction explaining difference of our approach compared to conventional equilibration papers. In particular how classicalization and markovianization go beyond the usual approach to thermalization.
- Added an extended discussion below what is now Eq. (29), comparing our bound to other (mostly numerical) results on more specific (and physical) models, detailing when the bound is useful.
- The proof Eq. (53) has been expanded with extra details.
- The title of the paper has changed slightly: "Non-Markovian" -> "Multitime"
- Changed paragraph 2 of the Introduction: "This IS a foundational question..."
- Changed paragraph 2 of the Introduction: "significantly interacting energy eigenstates" to “many significant populations in the energy eigenbasis….”
- Changed labelling of Hilbert spaces from numbers to letters on pages 2 and 3.
- "T" transpose notation clarifed with a comment in the second paragraph on page 3.
- A comment has been added to page 4 explaining trace properties of process tensor, " that tracing over a final
output leg of a process means..."
- Added sentence clarifying notation below Eq. (14).
- Page 5: "average indistinguishability" -> "time-average indistinguishability" .
- First sentence of Sec. III: dotted --> dashed.
- Added Eq. (23) and surrounding details.
- Added Eq. (28) and surrounding details.
- Added relevant citation to recent work Ref. [26]
- Updated penultimate paragraph of the conclusion to better detail comparisons with related work.

Published as SciPost Phys. Core 6, 043 (2023)

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